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If P has an integer root, so does f(P)

Source: India EGMO TST 2024/3

December 31, 2023
functionalgebrapolynomialEgmo tst

Problem Statement

Find all functions f:N↦Nf: \mathbb{N} \mapsto \mathbb{N} so that for any positive integer nn and finite sequence of positive integers a0,…,ana_0, \dots, a_n, whenever the polynomial a0+a1x+⋯+anxna_0+a_1x+\dots+a_nx^n has at least one integer root, so does f(a0)+f(a1)x+⋯+f(an)xn.f(a_0)+f(a_1)x+\dots+f(a_n)x^n.
Proposed by Sutanay Bhattacharya
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